If $$a, b, c, d$$   and $$p$$ are distinct real numbers such that $$\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left( {ab + bc + cd} \right)p + \left( {{b^2} + {c^2} + {d^2}} \right) \leqslant 0$$            then $$a, b, c, d$$   are in

Solution :
$$\eqalign{ & {\left( {ap - b} \right)^2} + {\left( {bp - c} \right)^2} + {\left( {cp - d} \right)^2} \leqslant 0 \cr & \Rightarrow \,\,ap - b = 0,bp - c = 0,cp - d = 0 \cr & \therefore \,\,p = \frac{b}{a} = \frac{c}{b} = \frac{d}{c}. \cr} $$